Standard Divergence in Manifold of Dual Affine Connections

Abstract

A divergence function defines a Riemannian metric G and dually coupled affine connections \(\left( \nabla , \nabla ^{*}\right) \) with respect to it in a manifold M. When M is dually flat, a canonical divergence is known, which is uniquely determined from \(\left\{ G, \nabla , \nabla ^{*}\right\} \). We search for a standard divergence for a general non-flat M. It is introduced by the magnitude of the inverse exponential map, where \(\alpha =-(1/3)\) connection plays a fundamental role. The standard divergence is different from the canonical divergence.